Download The Network Topology of the Interbank Market

The Network Topology of the Interbank Market
∗
Michael Boss1 , Helmut Elsinger2, Martin Summer1 , and Stefan Thurner3
1
Oesterreichische Nationalbank, Otto-Wagner-Platz 3, A-1011 Wien; Austria
Department of Finance, Universität Wien, Brünner Strasse 71, A-1210 Wien; Austria
3
Complex Systems Research Group, HNO, Universität Wien, Währinger Gürtel 18-20, A-1090; Austria
arXiv:cond-mat/0309582v1 25 Sep 2003
2
We provide an empirical analysis of the network structure of the Austrian interbank market based
on a unique data set of the Oesterreichische Nationalbank (OeNB). We show that the contract size
distribution follows a power law over more than 3 decades. By using a novel ”dissimilarity” measure
we find that the interbank network shows a community structure that exactly mirrors the regional
and sectoral organization of the actual Austrian banking system. The degree distribution of the
interbank network shows two different power law exponents which are one-to-one related to two
sub-network structures, differing in the degree of hierarchical organization. The banking network
moreover shares typical structural features known in numerous complex real world networks: a low
clustering coefficient and a relatively short average shortest path length. These empirical findings
are in marked contrast to interbank networks that have been analyzed in the theoretical economic
and econo-physics literature.
JEL: C73, G28
PACS: 89.65.Gh, 89.75.Hc, 89.65.-s,
Over the past years the physics community has largely
contributed to the analysis and to a functional understanding of the structure of complex real world networks. A key insight of this research has been the discovery of surprising structural similarities in very different networks, such as Internet, the World Wide Web,
collaboration networks, biological networks, communication networks, electronic circuits and power grids, industrial networks, nets of software components and energy
landscapes. See [1] for an overview. Remarkably, most
of these real world networks show degree distributions
which follow a power law, feature a certain pattern of
cliquishness, quantified by the clustering coefficient of
the network, and exhibit the so called small world phenomenon, meaning that the average shortest path between any two vertices (”degrees of separation”) in the
network can be surprisingly small [2]. Maybe one of the
most important contributions to recent network theory
is an interpretation of these network parameters with respect to stability, robustness, and efficiency of an underlying system, e.g. [3].
From this perspective financial networks seem a natural candidate to study. Indeed in the economic literature financial crises that have hit countries all around
the globe have led to a boom of papers on banking-crises,
financial risk-analysis and to numerous policy initiatives
to improve financial stability. One of the major concerns
in these debates is the danger of so called systemic risk:
∗ The
views expressed in this paper are stritcly the views of the
authors and do in no way commit the OeNB.
Correspondence to: Stefan Thurner; Complex Systems Research
Group, HNO, AKH-Wien, University of Vienna; Währinger Gürtel
18-20; A-1090 Vienna, Austria; Tel.: ++43 1 40400 2099; Fax:
++43 1 40400 3332; e-mail: [email protected]
the large scale breakdown of financial intermediation due
to domino effects of insolvency [4, 5]. The network of mutual credit relations between financial institutions is supposed to play a key role in the risk for contagious defaults.
Some authors in the theoretical economic literature on
contagion [6, 7, 8] suggest network topologies that might
be interesting to look at. In [6] it is suggested to study
a complete graph of mutual liabilities. The properties of
a banking system with this structure is then compared
to properties of systems with non complete networks. In
[7] a circular graph is contrasted with a complete graph.
In [8] a much richer set of different network structures is
studied. Yet, surprisingly little is known about the actual
empirical network topology of mutual credit relations between financial institutions. To our best knowledge the
network topology of interbank markets has so far not
been studied empirically.
The interbank network is characterized by the liability
(or exposure) matrix L. The entries Lij are the liabilities bank i has towards bank j. We use the convention
to write liabilities in the rows of L. If the matrix is read
column-wise (transposed matrix LT ) we see the claims or
interbank assets, banks hold with each other. Note, that
L is a square matrix but not necessarily symmetric. The
diagonal of L is zero, i.e. no bank self-interaction exists.
In the following we are looking for the bilateral liability matrix L of all (about N = 900) Austrian banks, the
Central Bank (OeNB) and an aggregated foreign banking
sector. Our data consists of 10 L matrices each representing liabilities for quarterly single month periods between
2000 and 2003. To obtain the Austrian interbank network from Central Bank data we draw upon two major
sources: we exploit structural features of the Austrian
bank balance sheet data base (MAUS) and the major
loan register (GKE) in combination with an estimation
technique.
2
N
X
Lij = bri
∀ i
and
N
X
Lij = bcj
∀ j
(1)
i=1
j=1
with r denoting row and c denoting column. Constraints
imposed by the knowledge about particular entries in Lij
are given by
bl ≤ Lij ≤ bu
for some i, j.
(2)
The aim is to find the matrix L (among all the matrices
fulfilling the constraints) that has the least discrepancy to
some a priori matrix U with respect to the (generalized)
cross entropy measure
N
N X
X
Lij
Lij ln
C(L, U ) =
.
(3)
Uij
i=1 j=1
(a)
8
10
slope= −1.87
6
10
histogram
The Austrian banking system has a sectoral organization due to historic reasons. Banks belong to one of
seven sectors: savings banks (S), Raiffeisen (agricultural)
banks (R), Volksbanken (VB), joint stock banks (JS),
state mortgage banks (SM), housing construction savings
and loan associations (HCL), and special purpose banks
(SP). Banks have to break down their balance sheet reports on claims and liabilities with other banks according
to the different banking sectors, Central Bank and foreign
banks. This practice of reporting on balance interbank
positions breaks the liability matrix L down to blocks
of sub-matrices for the individual sectors. The savings
banks and the Volksbanken sector are organized in a
two tier structure with a sectoral head institution. The
Raiffeisen sector is organized by a three tier structure,
with a head institution for every federal state of Austria.
The federal state head institutions have a central institution, Raiffeisenzentralbank (RZB) which is at the top of
the Raiffeisen structure. Banks with a head institution
have to disclose their positions with the head institution,
which gives additional information on L. Since many
banks in the system hold interbank liabilities only with
their head institutions, one can pin down many entries in
the L matrix exactly. This information is combined in a
next step with the data from the major loans register of
OeNB. This register contains all interbank loans above a
threshold of 360 000 Euro. This information provides us
with a set of constraints (inequalities) and zero restrictions for individual entries Lij . Up to this point one can
obtain about 90% of the L-matrix entries exactly.
For the rest we employ an estimation routine based
on local entropy maximization, which has already been
used to reconstruct unknown bilateral interbank exposures from aggregate information [9, 10]. The procedure
finds a matrix that fulfills all the known constraints and
treats all other parts (unknown entries in L) as contributing equally to the known row and column sums. These
sums are known since the total claims to other banks
have to reported to the Central Bank. The estimation
problem can be set up as follows: Assume we have a total of K constraints. The column and row constraints
take the form
4
10
2
10
0
10
0
10
1
10
2
contract10size L
3
10
4
10
ij
(b)
FIG. 1: The banking network of Austria (a). Clusters are
grouped (colored) according to regional and sectorial organization: R-sector with its federal state sub-structure: RB
yellow, RSt orange, light orange RK, gray RV, dark green
RT, black RN, light green RO, light yellow RS. VB-sector:
dark gray, S-sector: orange-brown, other: pink. Data is from
the September 2002 L matrix, which is representative for all
the other matrices. In (b) we show the contract size distribution within this network (histogram of all entries in L) which
follows a power law with exponent −1.87. Data is aggregated
from all 10 matrices.
U is the matrix which contains all known exact liability
entries. For those entries (bank pairs) ij where we have
no knowledge from Central Bank data, we set Uij = 1.
We use the convention that Lij = 0 whenever Uij = 0
and define 0 ln( 00 ) to be 0. This is a standard convex optimization problem, the necessary optimality conditions
can be solved efficiently by an algorithm described in
[11, 12]. As a result we obtain a rather precise (see below) picture of the interbank relations at a particular
point in time. Given L we plot the distribution (pdf)
of its entries in Fig. 1(b). The distribution of liabilities
3
tained in this way (Fig. 1a) can be compared to the
actual community structure in the real world. The result
for the community structure obtained from one representative data set is shown in Fig. 2. Results from other
datasets are practically identical. The algorithm identifies communities of banks which are coupled by a two or
three tier structure, i.e. the R, VB, and S sectors. For
banks which in reality are not structured in a hierarchical way, such as banks in the SP, JS, SM, HCL sectors,
no strong community structure is expected. By the algorithm these banks are grouped together in a cluster called
’other’. The Raiffeisen sector, with its substructure in
federal states, is further grouped into clusters which are
clearly identified as R banks within one of the eight federal states (B,St,K,V,T,N,O,S). In Fig. 2 these clusters
are marked as e.g. ’RS’, ’R’ indicating the Raiffeisen sector, and ’S’ the state of Salzburg. Overall, there were
31 mis-specifications into wrong clusters within the total
N = 883 banks, which is a mis-specification rate of 3.5 %,
demonstrating the quality of the dissimilarity algorithm
and - more importantly - the quality of the entropy approach to reconstruct matrix L.
2
DISSIMILARITY INDEX
follows a power law for more than three decades with an
exponent of −1.87, which is within a range which is well
known from wealth- or firm size distributions [13, 14].
To extract the network topology from these data, there
are three possible approaches to describe the structure as
a graph. The first approach is to look at the liability matrix as a directed graph. The vertices are all Austrian
banks. The Central Bank OeNB, and the aggregate foreign banking sector are represented by a single vertex
each. The set of all initial (starting) vertices is the set
of banks with liabilities in the interbank market; the set
of end vertices is the set of all banks that are claimants
in the interbank market. Therefore, each bank that has
liabilities with some other bank in the network is considered an initial vertex in the directed liability graph.
Each bank for which this liability constitutes a claim, i.e.
the bank acting as a counterparty, is considered an end
vertex in the directed liability graph. We call this representation the liability adjacency matrix and denote it by
Al (l indicating liability). Alij = 1 whenever a connection starts from row node i and leads to column node j,
and Alij = 0 otherwise. If we take the transpose of Al
we get the interbank asset matrix Aa = (Al )T . A second way to look at the graph is to ignore directions and
regard any two banks as connected if they have either a
liability or a claim against each other. This representation results in an undirected graph whose corresponding
adjacency matrix Aij = 1 whenever we observe an interbank liability or claim. Our third graph representation
is to define an undirected but weighted adjacency matrix
Aw
ij = Lij + Lji , which measures the gross interbank interaction, i.e. the total volume of liabilities and assets for
each node. Which representation to use depends on the
questions addressed to the network. For statistical descriptions of the network structure matrices A, Aa , and
Al will be sufficient, to reconstruct the community structure from a graph, the weighted adjacency matrix Aw
will be the more useful choice.
There exist various ways to find functional clusters
within a given network. Many algorithms take into account local information around a given vertex, such as
the number of nearest neighbors shared with other vertices, number of paths to other vertices, see e.g. [15, 16].
Recently a global algorithm was suggested which extends the concept of vertex betweenness [17] to links [18].
This elegant algorithm outperforms most traditional approaches in terms of mis-specifications of vertices to clusters, however it does not provide a measure for the differences of clusters. In [19] an algorithm was introduced
which - while having at least the same performance rates
as [18] - provides such a measure, the so-called dissimilarity index. The algorithm is based on a distance definition
presented in [20].
For analyzing our interbank network we apply this
latter algorithm to the weighted adjacency matrix Aw .
As the only preprocessing step we clip all entries in Aw
above a level of 300 m Euro for numerical reasons, i.e.
w
Aw
clip = min(A , 300m). The community structure ob-
1.5
1
0.5
0
3
5
7
9
11
13
15
16
18
19
20
CLUSTER #
42
95
55
24
88
100
135
61
69
75
139
SIZE
RB
RSt
RK
RV
RT
RN
RO
RS
VB
S
other SECTOR
FIG. 2: Community structure of the Austrian interbank market network from the September 2002 data. The dissimilarity
index is a measure of the ”differentness” of the clusters.
Degree Distributions: Like many real world networks,
the degree distribution of the interbank market follow
power laws for all three graphs Al , Aa , and A. Figure 3
(a) and (b) show the out-degree (liabilities) and in-degree
(assets) distribution of the vertices in the interbank liability network. Figure 3 (c) shows degree distribution of the
interbank connection graph A. In all three cases we find
two regions which can be fitted by a power. Accordingly,
we fit one regression line to the small degree distribution and one to the obvious power tails of the data using
an iteratively re-weighted least square algorithm. The
power decay exponents γtail to the tails of the degree
distributions are γtail (Al ) = 3.11, γtail (Aa ) = 1.73, and
γtail (A) = 2.01. The size of the out degree exponent is
within the range of several other complex networks, like
4
slope= −0.68613
3
histogram
10
2
10
1
10
(a)
slope= −3.1076
0
10 0
10
1
2
10
10
3
10
out degree
4
10
slope= −1.0831
3
10
2
10
.
(4)
It provides the probability that two vertices that are
connected to any given vertex are also connected with
one another. A high clustering coefficient means that
two banks that have interbank connections with a third
bank, have a greater probability to have interbank connections with one another, than will any two banks randomly chosen on the network. The clustering coefficient
is only well defined in undirected graphs. We find the
clustering coefficient of the connection network (A) to be
C = 0.12 ± 0.01 (mean and standard deviation over the
10 data sets) which is relatively small compared to other
networks. In the context of the interbank market a small
C is a reasonable result. While banks might be interested
in some diversification of interbank links keeping a link is
also costly. So if for instance two small banks have a link
with their head institution there is no reason for them to
additionally open a link among themselves.
Average Shortest Path Length: We calculate the average path length for the three networks Al , Aa , A with
the Dijkstra algorithm [29] and find an average path
length of ℓ̄(Al ) = ℓ̄(Aa ) = 2.59 ± 0.02. Note the possibility that in a directed graph not all nodes can be
reached and we restrict our statistics to the giant components of the directed graphs. The average path length
in the (undirected) interbank connection network A is
ℓ̄(A) = 2.26 ± 0.03. From these results the Austrian interbank network looks like a very small world with about
three degrees of separation. This result looks natural in
the light of the community structure described earlier.
The two and three tier organization with head institutions and sub-institutions apparently leads to short interbank distances via the upper tier of the banking system
and thus to a low degree of separation.
Our analysis provides a first picture of the interbank
1
10
(b)
slope= −1.7275
0
10 0
10
1
2
10
10
3
10
in degree
4
10
slope= −0.61557
3
10
histogram
3 × (number of triangles on the graph)
C=
number of connected triples of vertices
4
10
histogram
e.g. collaboration networks of actors (3.1) [21], sexual
contacts (3.4) [22]. Exponents in the range of 2 are for
example the Web (2.1) [23] or mathematicians’ collaboration networks (2.1) [24], and examples for exponents of
about 1.5 are email networks (1.5) [25] and co-autorships
(1.2) [26]. For the left part of the distribution (small
degrees) we find γsmall (Al ) = 0.69, γsmall (Aa ) = 1.01,
and γsmall (A) = 0.62. These exponents are small compared to other real world networks. Compare e.g. with
foodwebs (1) [27]. We have checked that the distribution for the low degrees is almost entirely dominated by
banks of the R sector. Typically in the R sector most
small agricultural banks have links to their federal state
head institution and very few contacts with other banks,
leading to a strong hierarchical structure, which is also
visible by plain eye in Fig. 1a. This hierarchical structure is perfectly reflected by the small scaling exponents
[28].
Clustering Coefficient: To quantify clustering phenomena within the banking network we use the so-called
clustering coefficient C, defined by
2
10
1
10
(c)
slope= −2.0109
0
10 0
10
1
10
2
degree
10
3
10
FIG. 3: Empirical out-degree (a) in-degree and (b) distribution of the interbank liability network. In (c) the degree
distribution of the interbank connection network is shown.
All the plots are histograms of aggregated data from all the
10 datasets.
network topology by studying a unique dataset for the
Austrian interbank market. Even though it is small the
Austrian interbank market is structurally very similar to
the interbank system in many European countries including the large economies of Germany, France, and Italy.
We show that the liability (contract) size distribution
5
follows a power law, which can be understood as being
driven by underlying size and wealth distributions of the
banks, which show similar power exponents. We find that
the interbank network shows – like many other realistic
networks – power law dependencies in the degree distributions. We could show that different scaling exponents
relate to different network structure in different banking
sectors within the total network. The scaling exponents
by the agricultural banks (R) are very low, due to the hierarchical structure of this sector, while the other banks
lead to scaling exponents also found in other complex
real world networks. Regardless of the size of the scaling exponent, the existence of a power law is a strong
indication for a stable network with respect to random
bank defaults or even intentional attack [3]. The interbank network shows a low clustering coefficient, a result
that mirrors the analysis of community structure which
shows a clear network pattern, where banks would first
have links with their head institution, whereas these few
head institutions hold links among each other. A consequence of this structure is that the interbank network is
a small world with a very low ”degree of separation” between any two nodes in the system. A further important
message of this work is that it allows to exclude large
classes of unrealistic types of networks for future modeling of interbank relations, which have so far been used in
the literature.
We thank J.D. Farmer for valuable comments to improve the paper and Haijun Zhou for making his dissimilarity index algorithm available to us. S.T. would like
to thank the SFI and in particular J.D. Farmer for their
great hospitality in the summer of 2003.
[1] S.N. Dorogovtsev and J.F.F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW,
(Oxford University Press, 2003).
[2] D.J. Watts, Small Worlds: The Dynamics of Networks
between Order and Randomness, (Princeton University
Press, 1999).
[3] R. Albert, H. Jeong, and A.-L. Barabasi, Nature 406,
378-382 (2000).
[4] P. Hartmann and O. DeBandt, European Central Bank
Working Paper Nr. 35 (2000).
[5] M. Summer, Open Economies Review 1, 43 (2003).
[6] F. Allen and D. Gale, Journal of Political Economy 108,
1 (2000).
[7] X. Freixas, L. Parigi, and J.C. Rochet, Journal of Money
Credit and Banking 32 (2000).
[8] S. Thurner, R. Hanel, S. Pichler, Quantitative Finance
3, 306-319 (2003).
[9] G. Sheldon and M. Maurer, Swiss Journal of Economics
and Statistics 134, 685 (1998).
[10] C. Upper and A. Worms, Deutsche Bundesbank, Dicussion paper 09 (2002).
[11] U. Blien and F. Graef, in I. Balderjahn, R. Mathar,
and M. Schader, Classification, Data Analysis, and Data
Highways (Springer Verlag, 1997).
[12] S.C. Fang, J.R. Rajasekra and J. Tsao, Entropy Optimization and Mathematical Programming (Kluwer Academic Publishers, 1997).
[13] S. Solomon, M. Levy, e-print: arXiv: cond-mat/0005416
(2000).
[14] R.L. Axtell, Science /bf 293 1818 (2001).
[15] S. Wasserman and K. Faust, Social Network Analysis:
Methods and Applications (Cambridge University Press,
1994).
[16] E. Ravasz, A.L. Somera, D.A. Mongru, Z.N. Oltvai, and
A.-L. Barabasi, Science 297, 1551 (2001).
[17] L.C. Freeman, Sociometry 40, 35 (1977).
[18] M. Girvan and M.E.J. Newman, Proc. Natl. Acad. Sci.
99, 7831 (2002).
[19] H. Zhou,
Phys. Rev. E (in print) e-print:
arXiv:cond-mat/0302030 (2003).
[20] H. Zhou, e-print: arXiv:physics/0302032 (2003).
[21] R. Albert, A.-L. Barabasi, Phys. Rev. Lett. 85, 52345237 (2000).
[22] F. Liljeros, C.F. Edling, L.A.N. Amaral, H.E. Stanley,
Y., Aberg, Nature 411, 907 (2001).
[23] R. Albert, H. Jeong, A.-L. Barabasi, Nature 401, 130
(1999).
[24] A.-L. Barabasi, H. Jeong, Z. Neda, E. Ravasz, A. Schubert, T. Vicsek, Physica A 311, 590 (2002).
[25] H. Ebel, L.I. Mielsch, S. Bornholdt, Phys. Rev. E 66,
036103 (2002).
[26] M.E.J. Newman, Phys. Rev. E 64, 016131 and 016132
(2001).
[27] J.M. Montoya, R.V. Sole, Santa Fe Institute working papers, 00-10-059 (2000).
[28] A. Trusina, S. Maslov, P. Minnhagen, K. Sneppen, eprint: arXiv:cond-mat/0308339 (2003).
[29] A. Gibbons, Algorithmic Graph Theory (Cambridge University Press, 1985).